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[[Digital signature]]s are a means to protect [[Digital data|digital information]] from intentional modification and to authenticate the source of digital information. [[Public key cryptography]] provides a rich set of different cryptographic algorithms the create digital signatures. However, the primary public key signatures currently in use ([[RSA (cryptosystem)|RSA]] and [[Elliptic Curve Digital Signature Algorithm|Elliptic Curve Signatures)]] will become completely insecure if scientists are ever able to build a moderately sized [[quantum computer]].<ref name=":2">{{Cite web|title = ETSI - Quantum-Safe Cryptography|url = http://www.etsi.org/technologies-clusters/technologies/quantum-safe-cryptography|website = ETSI|access-date = 2015-07-05|first = Sabine|last = Dahmen-Lhuissier}}</ref> [[Post-quantum cryptography|Post quantum cryptography]] is a class of cryptographic algorithms designed to be resistant to attack by a quantum cryptography. Several post quantum digital signature algorithms based on hard problems in lattices are being created replace the commonly used [[RSA (cryptosystem)|RSA]] and elliptic curve signatures. A subset of these lattice based scheme are based on a problem known as [[Ring learning with errors]]. Ring learning with errors based digital signatures are among the post quantum signatures with the smallest public key and signature sizes
== Background ==
Developments in [[quantum computing]] over the past decade and the optimistic prospects for real quantum computers within 20 years have begun to threaten the basic cryptography that secures the internet.<ref>{{Cite web|title = Quantum computing breakthrough claim from IBM|url = http://www.cio.co.uk/news/r-and-d/quantum-computing-breakthrough-claim-from-ibm-3609914/|access-date = 2015-06-01|first = Agam|last = Shah|archive-date = 2015-09-23|archive-url = https://web.archive.org/web/20150923203728/http://www.cio.co.uk/news/r-and-d/quantum-computing-breakthrough-claim-from-ibm-3609914/|url-status = dead}}</ref><ref>{{Cite news|title = Researchers Report Milestone in Developing Quantum Computer|url = https://www.nytimes.com/2015/03/05/science/quantum-computing-nature-google-uc-santa-barbara.html|newspaper = The New York Times|date = 2015-03-04|access-date = 2015-07-05|issn = 0362-4331|first = John|last = Markoff}}</ref> A relatively small [[quantum computer]] capable of processing only ten thousand of bits of information would easily break all of the widely used [[Public-key cryptography|public key]] cryptography algorithms used to protect privacy and digitally sign information on the internet.<ref name=":2" /><ref>{{Cite journal|title = Efficient Networks for Quantum Factoring|journal = Physical Review A|date = 1996|issn = 1050-2947|pages = 1034–1063|volume = 54|issue = 2|doi = 10.1103/PhysRevA.54.1034|pmid = 9913575|first1 = David|last1 = Beckman|first2 = Amalavoyal N.|last2 = Chari|first3 = Srikrishna|last3 = Devabhaktuni|first4 = John|last4 = Preskill|arxiv = quant-ph/9602016|bibcode = 1996PhRvA..54.1034B|s2cid = 2231795}}</ref>
One of the most widely used public key algorithm used to create [[digital signatures]] is known as [[RSA (cryptosystem)|RSA]]. Its security is based on the classical difficulty of factoring the product of two large and unknown primes into the constituent primes. The [[integer factorization problem]] is believed to be intractable on any conventional computer if the primes are chosen at random and are sufficiently large. However, to factor the product of two n-bit primes, a quantum computer with roughly 6n bits of logical [[qubit]] memory and capable of executing a program known as [[Shor's algorithm]] will easily accomplish the task.<ref>{{Cite journal|title = Oversimplifying quantum factoring|journal = Nature|date = July 11, 2013|issn = 0028-0836|pages = 163–165|volume = 499|issue = 7457|doi = 10.1038/nature12290|first1 = John A.|last1 = Smolin|first2 = Graeme|last2 = Smith|first3 = Alexander|last3 = Vargo|pmid=23846653|arxiv = 1301.7007|bibcode = 2013Natur.499..163S|s2cid = 4422110}}</ref> Shor's algorithm can also quickly break digital signatures based on what is known as the [[discrete logarithm]] problem and the more esoteric [[Elliptic curve cryptography|elliptic curve discrete logarithm]] problem. In effect, a relatively small quantum computer running Shor's algorithm could quickly break all of the digital signatures used to ensure the privacy and integrity of information on the internet today.
Even though we do not know when a quantum computer to break RSA and other digital signature algorithms will exist, there has been active research over the past decade to create cryptographic algorithms which remain secure even when an attacker has the resources of a quantum computer at their disposal.<ref name=":2" /><ref name=":4">{{Cite web|title = Introduction|url = http://pqcrypto.org/|website = pqcrypto.org|access-date = 2015-07-05}}</ref> This new area of cryptography is called [[Post-quantum cryptography|Post Quantum]] or [[Quantum Safe Cryptography|Quantum Safe]] cryptography.<ref name=":2" /><ref name=":4" /> This article is about one class of these algorithms: digital signatures based on the Ring Learning with Errors problem. The use of the general [[Learning with errors|Learning with Errors]] problem in cryptography was introduced by Oded Regev in 2005 and has been the source of several cryptographic designs.<ref>{{Cite web|title = The Learning with Errors Problem|url = http://www.cims.nyu.edu/~regev/papers/lwesurvey.pdf|website = www.cims.nyu.edu|access-date = 2015-05-24}}</ref>
The creators of the Ring-based Learning with Errors (RLWE) basis for cryptography believe that an important feature of these algorithms based on Ring-Learning with Errors is their provable reduction to known hard problems.<ref>{{Cite book |date = 2010|pages = 1–23|first1 = Vadim|last1 = Lyubashevsky|first2 = Chris|last2 = Peikert|first3 = Oded|last3 = Regev| title=Advances in Cryptology – EUROCRYPT 2010 | chapter=On Ideal Lattices and Learning with Errors over Rings | series=Lecture Notes in Computer Science | volume=6110 |citeseerx = 10.1.1.297.6108|doi=10.1007/978-3-642-13190-5_1| isbn=978-3-642-13189-9 |editor-last=Gilbert|editor-first=Henri }}</ref><ref>{{Cite web|title = What does GCHQ's "cautionary tale" mean for lattice cryptography?|url = http://www.cc.gatech.edu/~cpeikert/soliloquy.html|website = www.cc.gatech.edu|access-date = 2015-07-05|url-status = dead|archive-url = https://web.archive.org/web/20150706150530/http://www.cc.gatech.edu/~cpeikert/soliloquy.html|archive-date = 2015-07-06}}</ref> The signature described below has a provable reduction to the [[Shortest vector problem|Shortest Vector Problem]] in an [[Ideal lattice cryptography|ideal lattice]].<ref name=":0">{{Cite book|title=Cryptographic Hardware and Embedded Systems – CHES 2012|last1=Güneysu|first1=Tim|last2=Lyubashevsky|first2=Vadim|last3=Pöppelmann|first3=Thomas|date=2012|publisher=Springer Berlin Heidelberg|isbn=978-3-642-33026-1|editor-last=Prouff|editor-first=Emmanuel|series=Lecture Notes in Computer Science|volume=7428|pages=530–547|chapter=Practical Lattice-Based Cryptography: A Signature Scheme for Embedded Systems|doi=10.1007/978-3-642-33027-8_31|editor-last2=Schaumont|editor-first2=Patrick}}</ref> This means that if an attack can be found on the Ring-LWE [[cryptosystem]] then a whole class of presumed hard computational problems will have a solution.<ref>{{Cite journal|title = The shortest vector in a lattice is hard to approximate to within some constant|url = http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.7305|journal = In Proc. 39th Symposium on Foundations of Computer Science|date = 1998|pages = 92–98|first = Daniele|last = Micciancio}}</ref>
The first RLWE based signature was developed by Lyubashevsky in his paper "Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures"<ref name=":5">{{Cite book|publisher = Springer Berlin Heidelberg|date = 2009-01-01|isbn = 978-3-642-10365-0|pages = 598–616|series = Lecture Notes in Computer Science|first = Vadim|last = Lyubashevsky|editor-first = Mitsuru|editor-last = Matsui|doi = 10.1007/978-3-642-10366-7_35|title = Advances in Cryptology – ASIACRYPT 2009|volume = 5912|chapter = Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures}}</ref> and refined in "Lattice Signatures Without Trapdoors" in 2011.<ref name=":1">{{Cite journal|title = Lattice Signatures Without Trapdoors|url = http://eprint.iacr.org/2011/537|date = 2011|first = Vadim|last = Lyubashevsky| journal=Cryptology ePrint Archive }}</ref> A number of refinements and variants have followed. This article highlights the fundamental [[mathematical structure]] of RLWE signatures and follows the original Lyubashevsky work and the work of Guneysu, Lyubashevsky and Popplemann ([https://web.archive.org/web/20140518004537/http://www.di.ens.fr/~lyubash/papers/signaturechess.pdf GLP]).<ref name=":0" /> This presentation is based on a 2017 update to the GLP scheme called GLYPH.<ref name=":3">{{Cite web|url=https://eprint.iacr.org/2017/766.pdf|title=GLYPH: A New Instantiation of the GLP Digital Signature Scheme|last=Chopra|first=Arjun|date=2017|website=International Association of Cryptographic Research eprint Archive|archive-url=https://web.archive.org/web/20170828012937/https://eprint.iacr.org/2017/766.pdf|archive-date=28 August 2017|access-date=26 August 2017|url-status=bot: unknown}}</ref>
A RLWE-SIG works in the quotient [[ring of polynomials]] modulo a degree n polynomial Φ(x) with coefficients in the [[finite field]] Z<sub>q</sub> for an odd prime q ( i.e. the ring Z<sub>q</sub>[x]/Φ(x) ).<ref name=":1" /> Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod Φ(x). For this presentation a typical polynomial is expressed as:
<math>a(x) = a_0 + a_1x + a_{2}x^2 + \ldots + a_{n-3}x^{n-3} + a_{n-2}x^{n-2} + a_{n-1}x^{n-1}</math>
The field Z<sub>q</sub> has its representative elements in the set { -(q-1)/2, ...-1, 0, 1, ... (q-1)/2 }. When n is a power of 2, the polynomial Φ(x) will be the [[cyclotomic polynomial]] x<sup>n</sup> + 1. Other choices of n are possible but the corresponding cyclotomic polynomials are more complicated or their security not as well studied.
=== Generating "small" polynomials. ===
A RLWE signature uses polynomials which are considered "small" with respect to a measure called the "[[infinity norm]]". The [[infinity norm]] for a polynomial is simply the largest absolute value of the coefficients of the polynomial when those coefficients are viewed as integers in Z rather than Z<sub>q</sub> .<ref name=":0" /> The signature algorithm will create random polynomials which are small with respect to a particular infinity norm bound. This is easily done by randomly generating all of the [[Coefficient|coefficients of the polynomial]] (a<sub>0</sub>, ..., a<sub>n-1</sub>) in a manner which is guaranteed or very likely to be less than or equal to this bound. In the literature on Ring Learning with Errors, there are two common ways to do this:<ref name=":1" />
* Using [[Uniform distribution (discrete)|Uniform Sampling]] - The coefficients of the small polynomial are uniformly sampled from a set of small coefficients. Let b be an integer that is much less than q. If we randomly choose polynomial coefficients from the set: { -b, -b+1, -b+2. ... -2, -1, 0, 1, 2, ... , b-2, b-1, b} the infinity norm of the polynomial will be ≤ (b).
* Using Discrete Gaussian Sampling - For an odd integer q, the coefficients are randomly chosen by sampling from the set { -(q-1)/2 to (q-1)/2 } according to a discrete Gaussian distribution with mean 0 and distribution parameter σ. The references provide more details on this method.
In the RLWE signature GLYPH used as an example below, coefficients for the "small" polynomials will use the [[Uniform distribution (discrete)|uniform sampling]] method and the value b will be much smaller than the value q.<ref name=":0" />
=== Hashing to a "small" polynomial ===
Most RLWE signature algorithms also require the ability to [[Cryptographic hash function|cryptographically hash]] arbitrary bit strings into small polynomials according to some distribution. The example below uses a hash function, POLYHASH(ω), which accepts a bit string, ω, as input and outputs a polynomial with n coefficients such that exactly k of these coefficients have absolute value greater than zero and less than an integer bound b (see above).
=== Rejection
In the example which follows, the bound, '''β,''' will be (b - k), where b is the range of the uniform sampling described above and k will be the number of non-zero coefficients allowed in an "accepted" polynomial<ref name=":0" />
===
Following GLYPH and as noted above, the maximum degree of the polynomials will be n-1 and therefore have n coefficients.<ref name=":0" /> Typical values for n are 512, and 1024.<ref name=":0" /> The coefficients of these polynomials will be from the field F<sub>q</sub> where q is an odd prime congruent to 1 mod 4. For n=1024, GLYPH sets q = 59393, b=16383 and k the number of non-zero coefficients in the output of Polyhash equal to 16.<ref name=":3" /> The number of non-zero coefficients k produced by the hash function is equal to 32 for both cases.<ref name=":0" /> The security of the signature scheme is closely tied to the relative sizes of n, q, b, and k. Details on setting these parameters can be found in references 5 and 6 below.<ref name=":1" /><ref name=":0" /><ref name=":3" />
As noted above, the polynomial Φ(x) which defines the ring of polynomials used will be x<sup>n</sup> + 1. Finally, a(x) will be a randomly chosen and fixed polynomial with coefficients from the set { -(q-1)/2 to (q-1)/2 }. The polynomial a(x) should be chosen in a "[[Nothing up my sleeve number|Nothing Up My Sleeve]]" manner such as [[One-way function|one-way hashing]] the output of a true noise random number generator (TRNG) or using the digital expansion of well known mathematical constans such as pi or e. All signers and verifiers of signatures will know n, q, b, k, Φ(x), a(x) and '''β''' = b-k.
== Public key generation ==
An entity wishing to sign messages generates its public key through the following steps:
# Generate two small polynomials s(x) and e(x) with coefficients chosen uniformly from the set {-b,...-1, 0, 1, ..., b}
# Compute t(x) = a(x)·s(x) + e(x)
# Distribute t(x) as the entity's public key
The polynomials s(x) and e(x) serve as the private key and t(x) is the corresponding public key. The security of this signature scheme is based on the following problem. Given a polynomial t(x) find small polynomials f<sub>1</sub>(x) and f<sub>2</sub>(x) such that: a(x)·f<sub>1</sub>(x) + f<sub>2</sub>(x) = t(x)
If this problem is difficult to solve, then the signature scheme will be difficult to forge. [See the Wikipedia article on [[Ring Learning with Errors]] or [[Ideal lattice cryptography|Ideal Lattice Cryptography]] for more details on the theoretical difficulty of this problem]
== Signature generation ==
Following GLYPH,<ref name=":3" /> to sign a message m expressed as a bit string, the signing entity does the following:
# Generate two small polynomials y<sub>1</sub>(x) and y<sub>2</sub>(x) with coefficients from the set {-b, ..., 0, ...., b}
# Compute w(x) = a(x)·y<sub>1</sub>(x) + y<sub>2</sub>(x)
# Map w(x) into a bit string ω
# Compute c(x) = POLYHASH(ω | m) (This is a polynomial with k non-zero coefficients. The "|" denotes concatenation of strings)
# Compute z<sub>1</sub>(x) = s(x)·c(x) + y<sub>1</sub>(x)
# Compute z<sub>2</sub>(x) = e(x)·c(x) + y<sub>2</sub>(x)
# Until the infinity norms of z<sub>1</sub>(x) and z<sub>2</sub>(x) ≤ '''β = ('''B - k) go to step 1. (This is the rejection sampling step noted above)
# The signature is the triple of polynomials c(x), z<sub>1</sub>(x) and z<sub>2</sub>(x)
# Transmit the message along with c(x), z<sub>1</sub>(x) and z<sub>2</sub>(x) to the verifier.
== Signature Verification ==
Following GLYPH,<ref name=":3" /> to verify a message m expressed as a bit string, the verifying entity must possess the signer's public key (t(x)), the signature (c(x), z<sub>1</sub>(x), z<sub>2</sub>(x)), and the message m. The verifier does the following:
# Verify that the infinity norms of z<sub>1</sub>(x) and z<sub>2</sub>(x) ≤ '''β''' , if not reject the signature.
# Compute w'(x) = a(x)·z<sub>1</sub>(x) + z<sub>2</sub>(x) - t(x)c(x)
# Map w'(x) into a bit string ω'
# Compute c'(x) = POLYHASH(ω' | m)
# If c'(x) ≠ c(x) reject the signature, otherwise accept the signature as valid.
Notice that:
a(x)·z<sub>1</sub>(x) + z<sub>2</sub>(x) - t(x)c(x) = a(x)·[s(x)·c(x) + y<sub>1</sub>(x)] + z<sub>2</sub>(x) - [a(x)·s(x) + e(x)]c(x)
{{=}} a(x)·y<sub>1</sub>(x) + z<sub>2</sub>(x) - e(x)·c(x)
{{=}} a(x)y<sub>1</sub>(x) + e(x)·c(x) + y<sub>2</sub>(x) - e(x)·c(x)
{{=}} a(x)y<sub>1</sub>(x) + y<sub>2</sub>(x) = w(x) (as defined above)
This brief derivation demonstrates that the verification process will have c'(x) = c(x) if the signature was not tampered with.
== Further developments ==
The GLYPH signature scheme described in this document closely follows the work of Lyubashevsky, Gunesyu and Popplemen from 2011 and 2012. There are a number of other variations to their work. These include:
* Work by Bai and Galbraith on short signatures documented [https://eprint.iacr.org/2013/838 here].<ref>{{Cite web|title = Cryptology ePrint Archive: Report 2013/838|url = https://eprint.iacr.org/2013/838|website = eprint.iacr.org|access-date = 2016-01-17}}</ref>
* Work by Akleylek, Bindel, Buchmann, Kramer and Marson on security proofs for the signature with fewer security assumptions and documented [https://eprint.iacr.org/2015/755 here].<ref>{{Cite web|title = Cryptology ePrint Archive: Report 2015/755|url = https://eprint.iacr.org/2015/755|website = eprint.iacr.org|access-date = 2016-01-17}}</ref>
Another approach to signatures based on lattices over Rings is a variant of the patented NTRU family of lattice based cryptography. The primary example of this approach is a signature known as the Bimodal Lattice Signature Scheme (BLISS). It was developed by Ducas, Durmas, Lepoint and Lyubashevsky and documented in their paper "Lattice Signatures and Bimodal Gaussians."<ref>{{Cite web|title = Cryptology ePrint Archive: Report 2013/383|url = https://eprint.iacr.org/2013/383|website = eprint.iacr.org|access-date = 2016-01-17}}</ref> See [[BLISS signature scheme]]
{{Reflist}}
==External links==
{{ Cryptography navbox | public-key }}
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